Optimal L(3, 2, 1)-labeling of triangular lattice

An n-L(1,2,3)labeling of a simple graph G=(V,E) is a mapping f:V{0,1,,n} such that f(u)f(v)i when the distance between u and v is i for i=1,2,3. The L(1,2,3)labeling span(1,2,3)(G) of a graph G is the minimum n such that G admits an n-L(1,2,3) labeling. In this article, we prove a conjecture by Calamoneri (2013) by showing (3,2,1)(L6)=19 where L6 is the infinite triangular lattice. We also show that (4,2,1)(L6)=19 but (k,2,1)(L6)20 for all k5.

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